Insurance Coverage Dispute between the hospital and the insurnace

Send them both a letter, suggesting that since the services have been billed incorrectly, you're not liable for them either.  Suggest, ever so politely, that if they can't agree on that, then you'll have to ask for help from the state insurance department and the state board of medical examiners (or whoever regulates hospitals in your state, where I don't practice).  Suggest that you might want to talk to a lawyer, if this isn't fixed, about the failure of each of them to deal with you in good faith.  Suggest that the situation, in a formal complaint, is likely to make someone look very bad, and that someone won't be you.

And if that doesn't do it, by all means, hire a lawyer!